It is well known that a straightforward application of Grover's quantum search algorithm enables to solve SAT in O (2 n /2 ) steps. Ambainis (SIGACT News, 2004) observed that it is possible to use Grover's technique to similarly speed up a sophisticated algorithm for solving 3-SAT. In this note, we show that a similar speed up can be obtained for all major record-breaking algorithms for satisfiability. We also show that if we use Grover's technique only, then we cannot do better than quadratic speed up.
It is well known that a straightforward application of Grover's quantum search algorithm enables to solve SAT in O (2 n /2 ) steps. Ambainis (SIGACT News, 2004) observed that it is possible to use Grover's technique to similarly speed up a sophisticated algorithm for solving 3-SAT. In this note, we show that a similar speed up can be obtained for all major record-breaking algorithms for satisfiability. We also show that if we use Grover's technique only, then we cannot do better than quadratic speed up.
This study explores how well Grover's Algorithm performs in solving the Boolean Satisfiability Problem (SAT) using quantum circuits. The algorithm is implemented with IBM's Qiskit framework and compared to classical brute-force methods. Experiments focus on 3-SAT, 4-SAT, and 5-SAT problems, using quantum simulators and IBM quantum hardware. The results show that Grover's Algorithm is more efficient, offering a theoretical quadratic speedup over classical methods. However, practical issues like limited qubit availability, hardware noise, and optimization challenges impact its current performance. The data highlights the potential for quantum computing to scale and solve NP-complete problems. This research shows how quantum computing can improve problem-solving and lays the groundwork for future studies on more complex SAT problems and advanced quantum hardware.
This study explores how well Grover's Algorithm performs in solving the Boolean Satisfiability Problem (SAT) using quantum circuits. The algorithm is implemented with IBM's Qiskit framework and compared to classical brute-force methods. Experiments focus on 3-SAT, 4-SAT, and 5-SAT problems, using quantum simulators and IBM quantum hardware. The results show that Grover's Algorithm is more efficient, offering a theoretical quadratic speedup over classical methods. However, practical issues like limited qubit availability, hardware noise, and optimization challenges impact its current performance. The data highlights the potential for quantum computing to scale and solve NP-complete problems. This research shows how quantum computing can improve problem-solving and lays the groundwork for future studies on more complex SAT problems and advanced quantum hardware.
This paper's aim is to explore improvements to, and applications of, a fundamental quantum algorithm invented by Grover\cite{grover}. Grover's algorithm is a basic tool that can be applied to a large number of problems in computer science, creating quantum algorithms that are polynomially faster than fastest known and fastest possible classical algorithms that solve the same problems. Our goal in this paper is to make these techniques readily accessible to those without a strong background in quantum physics: we achieve this by providing a set of tools, each of which makes use of Grover's algorithm or similar techniques, which can be used as subroutines in many quantum algorithms.}{The tools we provide are carefully constructed: they are easy to use, and in many cases they are asymptotically faster than the best tools previously available. The tools we build on include algorithms by Boyer, Brassard, Hoyer and Tapp, Buhrman, Cleve, de Witt and Zalka and Durr and Hoyer.}{After creating our tools, we create several new quantum algorithms, each of which is faster than the fastest known deterministic classical algorithm that accomplishes the same aim, and some of which are faster than the fastest possible deterministic classical algorithm. These algorithms solve problems from the fields of graph theory and computational geometry, and some employ dynamic programming techniques. We discuss a breadth-first search that is faster than Θ(edges) (the classical limit) in a dense graph, maximum-points-on-a-line in O(N3/2lgN) (faster than the fastest classical algorithm known), as well as several other algorithms that are similarly illustrative of solutions in some class of problem. Through these new algorithms we illustrate the use of our tools, working to encourage their use and the study of quantum algorithms in general.
The K-SAT problem is a fundamental challenge in computational theory, widely used in artificial intelligence, cryptography, and optimization. Classical algorithms struggle with exponential time complexity, making them inefficient for large-scale instances. In this research, we propose an efficient quantum computing approach based on Grover’s algorithm to enhance the search process for satisfiable solutions in K-SAT problems. Our implementation leverages quantum superposition and amplitude amplification to explore multiple possible solutions simultaneously, reducing the search complexity from O (2ⁿ) in classical methods to O(√2ⁿ) in quantum computing. We design a quantum oracle that encodes the K-SAT clauses and integrates it into Grover’s iterative search framework. The performance is evaluated through Qiskit simulations, demonstrating a significant improvement in search efficiency compared to classical brute-force techniques. The results highlight the potential of quantum algorithms in solving complex combinatorial problems with enhanced speed and accuracy. This study contributes to the development of quantum-accelerated optimization methods, paving the way for real-world applications in machine learning, cryptanalysis, and large-scale data processing.
Satisfiability (SAT) is a central problem in computer science, and advances in SAT-solving algorithms have a far-reaching impact across many fields. Recent works have proposed quantum SAT solvers based on Grover's algorithm, a quantum search technique. However, Grover-based approaches face a key limitation: they typically require prior knowledge of the number of satisfying assignments of the target Boolean formula. This information is unavailable in most practical settings. Quantum counting can be used to estimate this quantity, but it incurs a computational overhead that is several orders of magnitude higher than Grover search. In this paper, we propose a novel quantum SAT solver based on entanglement and equivalence checking. Our method does not assume prior knowledge of the number of solutions and is computationally more efficient than quantum counting. Although the worst case time complexity is inevitably exponential, we prove that the expected time complexity of our approach is only constant time O(1) over random Boolean functions. Experimental results also support our theoretical claim.
The classical 3SUM conjecture states that the class of 3SUM-hard problems does not admit a truly subquadratic O(n2−δ)-time algorithm, where δ>0, in classical computing. The geometric 3SUM-hard problems have widely been studied in computational geometry and recently, these problems have been examined under the quantum computing model. For example, Ambainis and Larka [TQC'20] designed a quantum algorithm that can solve many geometric 3SUM-hard problems in O(n1+o(1))-time, whereas Buhrman [ITCS'22] investigated lower bounds under quantum 3SUM conjecture that claims there does not exist any sublinear O(n1−δ)-time quantum algorithm for the 3SUM problem. The main idea of Ambainis and Larka is to formulate a 3SUM-hard problem as a search problem, where one needs to find a point with a certain property over a set of regions determined by a line arrangement in the plane. The quantum speed-up then comes from the application of the well-known quantum search technique called Grover search over all regions. This paper further generalizes the technique of Ambainis and Larka for some 3SUM-hard problems when a solution may not necessarily correspond to a single point or the search regions do not immediately correspond to the subdivision determined by a line arrangement. Given a set of n points and a positive number q, we design O(n1+o(1))-time quantum algorithms to determine whether there exists a triangle among these points with an area at most q or a unit disk that contains at least q points. We also give an O(n1+o(1))-time quantum algorithm to determine whether a given set of intervals can be translated so that it becomes contained in another set of given intervals and discuss further generalizations.